Question

Using the vertex form, find a formula for the parabola with vertex (2,14) that passes through the point (1,7)?

7=a(1-2)^2+14
so I got a to be -7

so the answer should be y=-7(x-2)^2+14 but my online math hw says I'm not correct?

Answer 1

`y=-7(x-2)^2+14` or `x=-1/49(y-14)^2+2`

Explanation:

There can be two parabolas with vertex at `(2,14)` and passing through `(1,7)`.

One of the form `(y-14)=a(x-2)^2` and other `(x-2)=a(y-14)^2`

Case 1 - If `(y-14)=a(x-2)^2` passes through `(1,7)` then

`7-14=a(1-2)^2` i.e. `a=-7`

and equation is `(y-14)=-7(x-2)^2` i.e. `y=-7(x-2)^2+14`, a vertical parabola

Case 2 - If `(x-2)=a(y-14)^2` passes through `(1,7)` then

`1-2=a(7-14)^2` i.e. `49a=-1` i.e. `a=-1/49`

and equation is `x-2=-1/49(y-14)^2` i.e. `x=-1/49(y-14)^2+2`, a horizontal parabola

graph{(y+7x^2-28x+14)(49x+y^2-28y+98)((x-1)^2+(y-7)^2-0.05)((x-2)^2+(y-14)^2-0.05)=0 [-10.21, 9.79, 4.52, 14.52]}

Answer 2

`y=-7(x-2)^2+14`
`x=-1/49(y-14)^2+2`

Explanation:

Given -

Vertex `(2, 14)`
Passes through point `(1, 7)`

This parabola may be either open down or open to left.

The equation of the parabola that opens down.

`y=a(x-h)+k`

Where `(h, k)` is the vertex

`y=a(x-2)^2+14`

The parabola is passing through point `(1,7)`

Substitute this to find the value of `a`

`a(1-2)^2+14=7`
`a+14 = 7`

`a=7-14=-7`

Then the required equation is -

`y=-7(x-2)^2+14`

Parabola May open to left. Then the equation is -

`x=a(y-k)^2+h`

`x=a(y-14)^2+2`

`a(y-14)^2+2=x`

`a(7-14)^2+2=1`

`49a+2=1`

`49a=1-2=-1`

`a=-1/49`

Then the equation is -

`x=-1/49(x-14)^2+2`